16 BAIRE'S THEOREM AND ITS CONSEQUENCES 89
Now suppose that E, F are equipped with translation-invariant distances
d, d', respectively, compatible with their topologies (12.9.1). By (18.104.22.168), the
hypothesis that u(E) is not meager in F implies that, for each r > 0, there
exists p = p(r) > 0 such that B'(0; p) c «(B'(0; r)). By translation it follows
that for each x e E we have B'(u(x); p) c u(B'(x; r)). Hence it is enough to
prove the following lemma:
(22.214.171.124) Let E be a complete metric space, F a metric space, and u a
continuous mapping ofE into F with the following property: for each r > 0 there
exists p = p(r) > 0 such that B'(u(x); p) c u(B'(x; rj) for all xeE. Then
B'(u(x); p) c u(B'(x; 2r))for all xeE.
For each integer n ^ 1, there exists by hypothesis a number pn > 0 such
that B'(u(x); pn) c u(S'(x; 2~~n+1r)) for all x e E. We may take p1 = p and
(replacing pn if necessary by inf(pn, 2""")) assume that lim pn = 0. Let x0 be
any point of E, and let y e B'(u(x0); p). We shall prove that y e u(B'(x0; 2r)).
For this we define inductively a sequence COn^i of points of E such that,
for each /i^l, we have xneEf(xn^; 2~~n+1r) and u(xn) eB'(j; pn+1).
Suppose that xi9 ..., xn.1 have been chosen to satisfy these conditions. Then
we have y e B'Cwfo-i); P»), and since y(u(xn^)\ pn) c u(Bf(xH^i 2~n+1r))y
there exists a point xn e B'(^n-i; 2~w+1r) such that u(xn) e B'(y; pn+i)> So the
inductive construction can continue.
Now the sequence (xn) is a Cauchy sequence in E, because for each n ^ 0
d(xn, xn+p) £ 2-»r + 2""-^ + • • • + 2-"-*+1r ^ 2~M+1r
for all p > 0. Since E is complete, the sequence converges in E to a point
x such that d(x0, x) <J 2r. Since w is continuous, we have u(x) = lim W(A:W);
and since d'(u(xn), y) ^ pw+1, it follows that lim «(*„) = 7. Q.E.D.
This theorem has the following corollaries:
(12.16.9) Let E, F be two Fr£chet spaces. Then every continuous bijective linear
mapping ofE onto F is an isomorphism.
In this proposition it is essential to assume that both E and F are complete.
For example, if we take E = l£ (6.5) and F to be the canonical image of E
under the identity mapping E -» ^n(N) (7.1), endowed with the topology
induced by that of ^R(N), then this mapping is continuous because if y0 is an interior point of w(W), then so is -j>0, and